S S symmetry Article Space-Efficient Prime Knot 7-Mosaics Aaron Heap * and Natalie LaCourt SUNY Geneseo, Department of Mathematics, 1 College Circle, Geneseo, NY 14454, USA * Correspondence: [email protected] Received: 28 February 2020; Accepted: 30 March 2020; Published: 5 April 2020 Abstract: The concepts of tile number and space-efficiency for knot mosaics were first explored by Heap and Knowles in 2018, where they determined the possible tile numbers and space-efficient layouts for every prime knot with mosaic number 6 or less. In this paper, we extend those results to prime knots with mosaic number 7. Specifically, we find the possible values for the number of non-blank tiles used in a space-efficient 7 × 7 mosaic of a prime knot are 27, 29, 31, 32, 34, 36, 37, 39, and 41. We also provide the possible layouts for the mosaics that lead to these values. Finally, we determine which prime knots can be placed within the first of these layouts, resulting in a list of knots with mosaic number 7 and tile number 27. Keywords: knot theory; knot; mosaic; tile number; space efficient; mosaic number; quantum knot 1. Introduction Knot mosaics were first introduced by Lomonaco and Kauffman in [1] as a basic building block of blueprints for constructing an actual physical quantum system, with a mosaic knot representing a quantum knot. The mosaic system they developed consisted of creating a square array of tiles selected from the list of tiles given in Figure1. These mosaic tiles are identified, respectively, as T0, T1, T2, . ., T10. T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 Figure 1. Tiles used for constructing mosaic knots. The first mosaic tile, T0, is a blank tile, and the remaining mosaic tiles, referred to as non-blank tiles, depict pieces of curves that will by used to construct knots or links when appropriately connected. These non-blank tiles consist of single arcs, horizontal or vertical line segments, double arcs, and over/under knot projection crossings. A connection point of a tile is an endpoint of a curve drawn on the tile. A tile is suitably connected if each of its connection points touches a connection point of an adjacent tile. Definition 1. An n × n array of suitably connected tiles is called an n × n knot mosaic, or n-mosaic. Note that an n-mosaic could represent a knot or a link, as illustrated in Figure2. The first two mosaics depicted are 4-mosaics, and the third one is a 5-mosaic. In this paper, we will be working only with knots, not links. Symmetry 2020, 12, 576; doi:10.3390/sym12040576 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 576 2 of 17 Trefoil Knot Hopf Link Figure-8 Knot Figure 2. Examples of knot mosaics. In addition to the original eleven tiles T0 – T10, we will also make use of nondeterministic tiles, such as those in Figure3, when there are multiple options for a specific tile location. For example, if a tile location must contain a crossing tile T9 or T10 but we have not yet chosen which, we will use the nondeterministic crossing tile, shown as the first tile in Figure3. Similarly, if we know that a tile location must have four connection points but we do not know if the tile is a double arc tile (T7 or T8) or a crossing tile (T9 or T10), we will indicate this with a tile that has four connection points, as seen in the second tile of Figure3. If the tile contains dashed lines or arcs, these will indicate the options for that tile. The third tile in Figure3 could be a horizontal segment T5 or a single arc T2. Figure 3. Examples of nondeterministic tiles. There are a few knot invariants of primary importance in this paper. The crossing number of a knot is the least number of crossings in any projection of the knot. The remaining invariants are directly related to knot mosaics. The first is the mosaic number of a knot, introduced in [1]. Definition 2. The mosaic number of a knot K is defined to be the smallest integer m for which K can be represented on an m-mosaic. The next knot invariant is the tile number of a knot, introduced by Lee, Ludwig, Paat, and Peiffer in [2] and first explored by Heap and Knowles in [3]. Definition 3. The tile number of a knot K is the smallest number of non-blank tiles needed to construct K on any size mosaic. We say that a knot mosaic is minimal if it is a realization of the mosaic number of the knot depicted on it. That is, if a knot with mosaic number m is depicted on an m-mosaic, then that mosaic is a minimal knot mosaic. It turns out that the tile number of a knot may not be realizable on a minimal mosaic. This fact was discovered by Heap and Knowles in [4], where it was shown that the knot 910 has mosaic number 6 and tile number 27, but that on a 6-mosaic 32 non-blank tiles were required. The tile number 27 was only achievable on a larger mosaic. Because of this, it is also of some interest to know how many non-blank tiles are necessary to depict a knot on a minimal mosaic, which is known as the minimal mosaic tile number of a knot, first introduced in [3]. Definition 4. Let m be the mosaic number of K. The minimal mosaic tile number of K is the smallest number of non-blank tiles needed to construct K on an m-mosaic. So the knot 910 has mosaic number 6, tile number 27, and minimal mosaic tile number 32, with the tile number achieved on a 7-mosaic. 910 is the simplest knot for which the tile number and minimal mosaic tile number are not equal. In this paper, we give more examples of this phenomenon. As we work with knot mosaics, we can move parts of the knot around within the mosaic via mosaic planar isotopy moves to obtain a different knot mosaic that depicts the same knot. Two knot mosaic diagrams are of the same knot type (or equivalent) if we can change one to the other via a sequence of Symmetry 2020, 12, 576 3 of 17 these mosaic planar isotopy moves. An examples of a mosaic planar isotopy move is given in Figure4, which is equivalent to a Reidemeister Type I move. If we have a mosaic that has one of these 2 × 2 submosaics within it, then that submosaic can be replaced by either of the other two without changing the knot type of the depicted knot. While these moves are technically tile replacements within the mosaic, they are analogous to the planar isotopy moves used to deform standard knot diagrams. A more complete list of these moves are given and discussed in [1,5]. We will make significant use of these moves throughout this paper, as we attempt to construct knot mosaics that use the least number of non-blank tiles. Figure 4. Example of a mosaic planar isotopy move. A knot mosaic is called reduced if there are no reducible crossings in the knot mosaic diagram. A crossing in a knot diagram is reducible if there is a circle in the projection plane that meets the diagram transversely at the crossing but does not meet the diagram at any other point. Removable crossings are unnecessary and can be easily removed by twisting. One such reducible crossing is given in the first 2 × 2 submosaic of Figure4. Another example is given in Figure5, together with a transverse circle, where the crossing in the fourth row and third column is reducible. If we want to create knot mosaics efficiently, using the least number of non-blank tiles necessary, we will want to avoid these reducible crossings. Figure 5. Reducing a reducible knot mosaic. Definition 5. A knot n-mosaic is space-efficient if it is reduced and the number of non-blank tiles is as small as possible on an n-mosaic without changing the knot type of the depicted knot. The number of non-blank tiles in a knot mosaic that is space-efficient cannot be decreased through a sequence of mosaic planar isotopy moves. In Figure6, the two knot mosaics depict the same knot (the 51 knot). However, the first knot mosaic uses nineteen non-blank tiles and the second knot mosaic uses only seventeen. In fact, seventeen is the minimum number of non-blank tiles possible to create this knot on a 5-mosaic. Therefore, the second mosaic is space-efficient, but the first one is not. Figure 6. Space-inefficient and space-efficient mosaics of the 51 knot. In [3], the possible layouts for space-efficient n-mosaics, together with the possible values of the minimal mosaic tile numbers and tile numbers, are given for all n ≤ 6. In the supplement to [4], we are provided with a table of knot mosaics that includes space-efficient mosaics for all prime knots with mosaic number 6 or less. In each of these prime knot mosaics, either the tile number or minimal mosaic tile number is realized. In this paper, we expand upon these ideas to include 7-mosaics. Symmetry 2020, 12, 576 4 of 17 For a quality introduction to knot mosaics, we refer the reader to [2]. For more details related to traditional knot theory, we refer the reader to [6] by Adams. We also point out that throughout this paper we make use of KnotScape [7], created by Thistlethwaite and Hoste, to verify that a given knot mosaic represents the specific knot we claim it does.